You are provided with 300.0 mL of a buffer solution consisting of 0.200 M H3BO3 and 0.250 M NaH2BO3.

a. Calculate the pH of the buffer solution.
b. Calculate the pH of the buffer solution if 1.0 mL of 6.0 M HCl was added to it.
c. Calculate the volume of 6.0 M NaOH that must be added to raise the pH of the buffer solution to 10.00.

a. 9.34

b. 9.06

c. 6  mL

Explanation:

Part a.

The pH of a buffer  solution is given by the Henderson-Hasselbach equation:

pH = pKa + log [A⁻] / [HA]

where pKa is the negative log of Ka for the weak acid H₃BO₃  and can be obtained from reference tables, [A⁻] and [HA] are the concentrations of the weak conjugate base H₂BO₃⁻ and and the weak acid H₃BO₃ respectively.

Proceeding with the calculations, we have

Ka H₃BO₃ = 5.80 x 10⁻¹⁰

pKa = - log (5.80 x 10⁻¹⁰) = 9.24

[H₂BO₃⁻ ] = 0.250 M

[H₃BO₃] = 0.200 M

pH = 9.24 + log (0.250/0.200) = 9.34

part b.

When 1.0 mL of 6.0 M HCl is added to the buffer , we know that it will react with the conjugate base in the buffer doing what buffers do: keeping the pH within a small range according to the capacity of the buffer:

H₂BO₃⁻ + H⁺ ⇒ H₃BO₃

So lets calculate the new concentrations of acid and conjugate base after reaction and apply the Henderson equation again:

Initial # of moles:

H₃BO₃  = 0.300 L x 0.200 mol/L = 0.06 mol

H₂BO₃⁻ = 0.200 L x 0.250 mol/L = 0.05 mol

mol HCl = 0.001 L x 6.0 mol/L = 0.006 mol

After reaction

H₃BO₃ = 0.06 mol + 0.006 mol = 0.066 mol

H₂BO₃⁻ = 0.05 mol - 0.006 mol = 0.044 mol

New pH

pH = 9.24 + log ( 0.044 / 0.66 ) = 9.06

Note: There is no need to calculate the new concentrations since we have a quotient in the expression where the volumes cancel each other.

Part c.

We will be using the Henderson-Hasselbach equationm again but now to calculate ratio [H₂BO₃⁻] / [HBO₃] that will give us a pH of 10.00. Thenwe will  make use of the stoichiometry of the reaction to calculate the volume of NaOH required.

pH =    pKa + log[H₂BO₃⁻]-[H₃BO₃]

10.00 = 9.24 + log [H₂BO₃⁻]-[H₃BO₃]

⇒[H₂BO₃⁻] / [H₃BO₃] = antilog (0.76) = 5.75

Initiall # moles:

mol H₃BO₃ = 0.06 mol

mol H₂BO₃ = 0.05 mol

after consumption of H₃BO₃ from the reaction with NaOH:

H₃BO₃ + NaOH ⇒ Na⁺ + H₂BO₃⁻ + H₂O

mol H₃BO₃ = 0.06 - x

mol H₂BO₃⁻ = 0.05+ x mol

Therefore we have the algebraic expression:

[H₂BO₃⁻] / [H₃BO₃] = mol H₂BO₃⁻ / mol HBO₃ = 5.75

( again volumes cancel each other)

0.05 + x / 0.06 - x = 5.75 ⇒ x =  0.044

SO 0.037 mol NaOH were required, and since we know Molarity = mol / V we can calculate the volume of 6.0 M NaOH added:

V = 0.044 mol / 6.0 mol/L = 0.0073 L

V = 7.3 mL

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